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That should be an easy question to answer with proper preparation! If you
have already been a project manager, then review your projects; select the
most challenging one; and prepare an answer that you can deliver
comfortably and confidently!

The most challenging project has time, cost or quality constraints. All or a combination of these pose challenges. The best way to tackle this is by effective planning and simulation. Periodical tracking and problem resolution is an absolute must.

The idea behind asking this question is to judge the depth of the knowledge a person is having. Another is how mature you are in taking the challenges. First one is for less experienced candidates and latter one is for PM.

Pick a project which was personally challenging for you and where you made a personal contribution in resolving the challenges. Focus on your contribution not generics.
The employer is looking for problem solving initiatives.

If I were hiring a project manager on my team, I would be looking at how the candidate responded to the challenge based on the "complexity" of the 'most challenging project' - what was his/her project management approach, attitude, competency, how he/she adapted the situation, whether he/she maintained his/her integrity while doing so.

1. Challenge for you i.e. you learnt the most, worked in a new scenario,
etc. If you already knew what all has to be done in the project then the
project was a easy one for you not a challenging for you.

2. From Company's (or project's) perspective. Where you have put your
maximum effort like, brought a dying project back on track, changed the
way as to how it was being handled, etc.

It is the not the answer. They are looking for your capability to analyze an issue and reach a solution.

Even trade-off is not a bad thing all the time. Sometimes there will be only one (trade off) option to solve a problem and the employer may want to know are you mature enough to accept the trade-off to resolve the issue or give importance to your ego first - which will complicate the issue further.

I agree with Satish - when I'm interviewing a PM who doesn't talk about how s/he handled people issues on their project(s), then I'm leery of their truthfulness, or their willingness to talk/work through difficulties. Everyone from executives through team members, is a stakeholder on the project. You WILL have people issues as everyone has a different agenda. I want to know how you manage your stakeholders as well as how well you can handle process and technical issues.

IMHO its a question to assess a person's ability to understand the project constraints and how objectively one looks up to a project challenge. The interviewer also tries to understand how the candidate attacks each constraint within the realm of project management principles and sticks to the basics unless out of box thinking is required. This is one of the most common questions for wannabe project managers.

It's all about classic PM skills revolving around cost, schedule, and quality on delivering results as manifested through the challenges presented by resources (mostly people), stakeholder expectations and behavior, the management of issues and risk, and how well you, as a PM, communicate!. A response should center around how well one performs the classic Plan-Do-Check-Act (PDCA) cycle. You have your plans and contingencies, you execute plans, check on their efficacy, take appropriate action, either proactively or reactively, and repeat the cycle throughout the project.

Next look at your personal portfolio of projects managed and apply this lens to each project to see the commonalities of response or improving changes you made based on the experience each project yielded. Now you're ready to answer the interview question.